It should be noted that there is a second resonant contribution to the flavonstrahlung cross-section: in addition to \( p p \rightarrow {Z^\prime }^\ast \rightarrow Z^\prime \vartheta \)  (case 1), discussed above, the cross-section also picks up a contribution from \( p p \rightarrow Z^\prime \rightarrow {Z^\prime }^\ast \vartheta \)  (case 2), where the intermediate \( Z^\prime \)  is on-shell. Choosing for concreteness the leading decay channels \( Z^\prime \rightarrow \mu ^- \mu ^+ \)  and \( \vartheta \rightarrow W^- W^+ \) , case 2 yields a resonance peak in the invariant mass of the 4-particle final state at \( q^2~=~M_{Z^\prime }^2 \)  as opposed to the di-muon resonance in case 1. We use the event generator MadGraph5_aMC@NLO v.3.4.1 [1]} (abbreviated “MG5_aMC” in the following) to illustrate the two resonances schematically in figure REF  for the final state \( \mu ^+ \mu ^- W^+ W^- \)  at a hadron collider. The orange and red lines corresponds to cases 1 and 2, respectively. We have used the condition \( | M^\ast - M | < 5 \Gamma \)  as the definition of a propagator being on-shell, with \( M^\ast \)  the invariant mass of the four-momentum carried by the propagator and \( M \)  and \( \Gamma \)  the pole mass and width of the particle. This is achieved using the \( \text{BW}_\text{cut} \)  parameter in MG5_aMC.
